21 solved GATE DA questions on Linear Algebra, drawn from 3 exam years and grouped by year. Every question shows the official answer and a step-by-step solution.
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Revision companion
Track dimensions and interpret calculations as statements about linear maps. Elimination describes solvability, projections describe approximation, and eigenvectors describe invariant directions. These selected revision notes emphasize checks that expose errors before they propagate into larger problems.
Our study notes and original examples support the PYQs below; they are not official exam questions or a replacement for the current syllabus.
An m × n matrix maps n-dimensional inputs to m-dimensional outputs. Row operations on the augmented matrix preserve solutions. Ax = b is consistent exactly when rank(A) equals rank([A | b]); a pivot in the augmented column signals inconsistency.
Check yourself: Did you apply each row operation to the right-hand side too?
A basis must span its space and be independent. Rank counts independent columns, equivalently pivots. For n columns, rank plus nullity equals n. Echelon-form pivot positions identify basis columns for the column space; select those columns from the original matrix.
Check yourself: Does your null-space dimension count the free input coordinates?
Projection onto a nonzero vector u is u(uᵀb)/(uᵀu). The residual is orthogonal to u. More generally, least-squares solutions satisfy Aᵀ(Ax − b) = 0. The familiar inverse formula requires independent columns; a singular AᵀA does not mean no minimizing solution exists.
Check yourself: Is the residual orthogonal to every column of A?
Solve det(A − λI) = 0 for eigenvalues, then solve (A − λI)v = 0 for nonzero eigenvectors. Their sum and product, counting multiplicities, equal trace and determinant. Real symmetric matrices admit an orthonormal eigenbasis; repeated eigenvalues do not guarantee this for arbitrary matrices.
Check yourself: Can you verify each proposed pair directly through Av = λv?
An SVD writes A = UΣVᵀ with orthonormal singular directions. Singular values are nonnegative square roots of eigenvalues of AᵀA; their nonzero count equals rank. Keeping the largest k terms gives a best rank-at-most-k approximation in Frobenius norm, linking matrix structure to compression.
Check yourself: Are you confusing singular values with potentially negative eigenvalues?
Original teaching example · not a PYQ
For A = [[2, 1], [1, 2]], find its eigenvalues, one eigenvector for each, and its rank. Use trace and determinant as independent checks.
Eigenvalues: 3 and 1; corresponding eigenvectors: (1, 1)ᵀ and (1, −1)ᵀ. Rank: 2.
For B = [[1, 2], [2, 4]] and b = (3, 6)ᵀ, find rank(B), nullity(B), and all solutions of Bx = b.
Rank = 1; nullity = 1; x = (3 − 2t, t)ᵀ for every real t.
The second augmented row is twice the first, leaving one pivot: x₁ + 2x₂ = 3. Two columns minus one pivot gives nullity one. Set x₂ = t to parameterize every solution; t = 0 gives (3, 0)ᵀ.
Supplemental reading, not an official GATE reading list or an endorsement of these notes.
Free supplemental lectures and problem sets by Gilbert Strang, with solutions for checking work on systems, vector spaces, projections, and eigenvalues.
This page shows 21 recent questions from the released archive, newest first. For older questions and complete papers, browse all GATE DA papers. Questions can carry more than one subject tag; counts are not marks weightage.
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